We proved here that \(f\) right-sided and left-sided limits vanish at all points. Therefore \(\limsup\limits_{x \to a} f(x) \le f(a)\) at every point \(a\) which proves that \(f\) is upper semi-continuous on \(\mathbb R\). However \(f\) is continuous at all \(a \in \mathbb R \setminus \mathbb Q\) and discontinuous at all \(a \in \mathbb Q\).

The fundamental theorem of calculus asserts that for a continuous real-valued function \(f\) defined on a closed interval \([a,b]\), the function \(F\) defined for all \(x \in [a,b]\) by
\[F(x)=\int _{a}^{x}\!f(t)\,dt\] is uniformly continuous on \([a,b]\), differentiable on the open interval \((a,b)\) and \[
F^\prime(x) = f(x)\]
for all \(x \in (a,b)\).

The converse of fundamental theorem of calculus is not true as we see below.

\(\sum u_n\) is a positive convergent series then \((\sqrt[n]{u_n})\) is bounded?

Is true. If not, there would be a subsequence \((u_{\phi(n)})\) such that \(\sqrt[\phi(n)]{u_{\phi(n)}} \ge 2\). Which means \(u_{\phi(n)} \ge 2^{\phi(n)}\) for all \(n \in \mathbb N\) and implies that the sequence \((u_n)\) is unbounded. In contradiction with the convergence of the series \(\sum u_n\).

If \((u_n)\) is non-increasing and converges to zero then \(\sum u_n\) converges?

Is not true. A famous counterexample is the harmonic series \(\sum \frac{1}{n}\) which doesn’t converge as \[
\displaystyle \sum_{k=p+1}^{2p} \frac{1}{k} \ge \sum_{k=p+1}^{2p} \frac{1}{2p} = 1/2,\] for all \(p \in \mathbb N\).

We build in this article a strictly increasing continuous function \(f\) that is differentiable at no point of a null set \(E\). The null set \(E\) can be chosen arbitrarily. In particular it can have the cardinality of the continuum like the Cantor null set.

Conclusion

Gabriel’s horn could be filled with a finite quantity of paint… therefore painting a surface with infinite area. Unfortunately the thickness of the paint coat is converging to \(0\) as \(z\) goes to \(\infty\), leading to a paint which won’t be too visible!